We consider rupture initiation and instability on a displacement-weakening interface. It is assumed to follow a power-law relation between a component of displacement discontinuity (whether tensile opening in mode I or shear slippage in modes II or III) and the reduction from peak strength of a corresponding component of stress (normal or shear stress) on the interface. That is, the stress decrease from peak strength, as the interface discontinuity develops, is assumed to be proportional to displacement-discontinuity to some exponent \(n > 0\). The study is done in the 2D context of plane or anti-plane strain, for an initially coherent interface which is subjected to a locally peaked “loading” stress which increases quasi-statically in time. We seek to establish the instability point, when no further quasi-static solution exists for growth of the ruptured zone along the interface, so that dynamic rupture ensues. We have previously addressed the case of linear displacement-weakening \((n = 1)\), and proven the remarkable result that for an unbounded solid, the length of the displacement-weakening zone along the interface at instability is universal, in the sense of being independent of the detailed spatial distribution of the locally peaked loading stress. Present results show that such universality does not apply when \(n\) differs from \(1\). Also, if \(n < 2/3\), there is no phase of initially quasi-static enlargement of the rupturing zone; instead instability will occur as soon as the maximum value of the loading stress reaches the peak strength. We first employ an energy approach to give a Rayleigh–Ritz approximation for the dependence of quasi-static rupture length and maximum displacement-discontinuity on the loading stress distribution of a quadratic form. Results, depending on curvature of the loading distribution, show that qualitative features of the displacement-discontinuity development are significantly controlled by n, with the transition noted at \(n = 2/3\). Predictions of the simple energy approach are in reasonable quantitative agreement with full numerical solutions and give qualitative features correctly.